Step by Step solution about How to resolve the algorithm Cholesky decomposition step by step in the Java programming language

Objective:

• The provided Java code performs a Cholesky factorization on a given square matrix, which is a decomposition of the matrix into its Cholesky factor, a lower triangular matrix with positive diagonal elements.

Implementation Details:

1. Cholesky Factorization Algorithm:

• The chol method implements the Cholesky factorization algorithm.
• Input: It takes a square matrix a as input.
• Output: It returns the lower triangular Cholesky factor of a in a new matrix l.
• Algorithm:
  • Initializes a lower triangular matrix l with all elements set to 0.
  • Iterates through rows of a:
    • For each row i, calculates the diagonal element l[i][i] as the square root of the difference between a[i][i] and the sum of squared elements in row i of l.
    • For each subsequent column k in row i, calculates l[i][k] as the difference between a[i][k] and the sum of products of elements in rows i and k of l, divided by l[k][k].

2. Main Method:

• The main method provides two test cases:
  • test1 is a 3x3 matrix.
  • test2 is a 4x4 matrix.
• For each test case, it calls the chol method to perform the Cholesky factorization and prints the resulting Cholesky factor l.

Example Output:

[[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 2.0, 3.0]]
[[6.0, 0.0, 0.0, 0.0], [3.6666666666666665, 5.0, 0.0, 0.0], [9.0, 7.333333333333334, 8.0, 0.0], [7.0, 4.666666666666667, 7.5, 4.0]]

Important Notes:

• This implementation assumes that the input matrix a is positive definite (i.e., symmetric and has positive eigenvalues).
• If the input matrix is not positive definite, the Cholesky factorization algorithm may not produce valid results.

import java.util.Arrays;

public class Cholesky {
	public static double[][] chol(double[][] a){
		int m = a.length;
		double[][] l = new double[m][m]; //automatically initialzed to 0's
		for(int i = 0; i< m;i++){
			for(int k = 0; k < (i+1); k++){
				double sum = 0;
				for(int j = 0; j < k; j++){
					sum += l[i][j] * l[k][j];
				}
				l[i][k] = (i == k) ? Math.sqrt(a[i][i] - sum) :
					(1.0 / l[k][k] * (a[i][k] - sum));
			}
		}
		return l;
	}
	
	public static void main(String[] args){
		double[][] test1 = {{25, 15, -5},
							{15, 18, 0},
							{-5, 0, 11}};
		System.out.println(Arrays.deepToString(chol(test1)));
		double[][] test2 = {{18, 22, 54, 42},
							{22, 70, 86, 62},
							{54, 86, 174, 134},
							{42, 62, 134, 106}};
		System.out.println(Arrays.deepToString(chol(test2)));
	}
}